Topological error correction with a Gaussian cluster state

Topological error correction provides an effective method to correct errors in quantum computation. It allows quantum computation to be implemented with a higher error threshold and high tolerating loss rates. We present a topological error correction scheme with continuous variables based on an eight-partite Gaussian cluster state. We show that topological quantum correlation between two modes can be protected against a single quadrature phase displacement error occurring on any mode and some of two errors occurring on two modes. More interestingly, some cases of errors occurring on three modes can also be recognized and corrected, which is different from the topological error correction with discrete variables. We show that the final error rate after correction can be further reduced if the modes are subjected to identical errors occurring on all modes with equal probability. The presented results provide a feasible scheme for topological error correction with continuous variables and it can be experimentally demonstrated with a Gaussian cluster state.

Figure 1

(a) The graph structure of the topological eight-partite CV cluster state. The connected lines represent the interaction between the neighbor nodes. Each node stands for an optical mode. (b) The beam-splitter network for the preparation of the cluster state. The ellipsoids represent the input squeezing states. The spheres represent the output cluster states. Circles including $i$ are Fourier transforms and $-1$ is a ${180}^{\circ }$ rotation in the phase space.

Figure 2

The circuit for error syndrome recognition and correction for the CV TEC. Here, we take the case of an error that occurs on mode ${\widehat{C}}_5$ as an example.

Figure 3

The dependence of the quantum correlation of ${\widehat{p}}_5-{\widehat{p}}_6$ on the squeezing of the initial squeezed states for different cases. Curve a is the corresponding shot-noise level for the protected correlation. Curve b is the correlation variance when no correction is used. Curve c is the correlation variance when no error has occurred. Curve d is the correlation variance when the TEC is used.

Figure 4

The dependence of the error rates for the protected quantum correlation on the single-mode error rate $p$. Curve a is the error rate for the protected correlation without TEC. Curve b is the error rate of TEC for the qubit. Curve c is the error rate of the CV TEC.